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applied cryptography - protocols, algorithms, and source code in c

# Xor m one bit at a time with the output of the

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Unformatted text preview: Keyword Brief Full Advanced Search Search Tips (Publisher: John Wiley & Sons, Inc.) Author(s): Bruce Schneier ISBN: 0471128457 Publication Date: 01/01/96 Search this book: Go! Previous Table of Contents Next ----------- Carol’s public key is e, her private key is d, and the RSA modulus is n. (1) Alice and Bob agree on a random k and an m such that km a e (mod n) They should choose the numbers randomly, using a coin-flip protocol to generate k and then computing m. If both k and m are greater than 3, the protocol continues. Otherwise, they choose again. (2) Alice and Bob generate a random ciphertext, C. Again, they should use a coin-flip protocol. (3) Alice, using Carol’s private key, computes M = Cd mod n She then computes X = Mk mod n and sends X to Bob. (4) Bob confirms that Xm mod n = C. If it does, he believes Alice. A similar protocol can be used to demonstrate the ability to break a discrete logarithm problem [888]. Zero-Knowledge Proof that n Is a Blum Integer There are no known truly practical zero-knowledge proofs that n =pq, where p and q are primes congruent to 3 modulo 4. However, if you allow n to be of the form prqs, where r and s are odd, then the properties which make Blum integers useful in cryptography still hold. And there exists a zero-knowledge proof that n is of that form. Assume Alice knows the factorization of the Blum integer n, where n is of the form previously discussed. Here’s how she can prove to Bob that n is of that form [660]. (1) Alice sends Bob a number u which has a Jacobi symbol -1 modulo n. (2) Alice and Bob jointly agree on random bits: b1 , b2 , ..., bk. (3) Alice and Bob jointly agree on random numbers: x1 , x2 , ..., xk. (4) For each i = 1, 2,..., k, Alice sends Bob a square root modulo n, of one of the four numbers: xi, -xi, uxi, -uxi. The square root must have the Jacobi symbol bi. The odds of Alice successfully cheating are one in 2k. 23.12 Blind Signatures The notion of blind signatures (see Section 5.3) was invented by David Chaum [317,323], who also invented their firs...
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